A rational splitting of a based mapping space

Abstract

Let ${\mathcal{ℱ}}_{\ast }\left(X,Y\right)$ be the space of base-point-preserving maps from a connected finite CW complex $X$ to a connected space $Y$. Consider a CW complex of the form $X{\cup }_{\alpha }{e}^{k+1}$ and a space $Y$ whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map $\alpha :S^k\to X$ is greater than the Whitehead length $WL\left(Y\right)$ of $Y$, then ${\mathcal{ℱ}}_{\ast }\left(X{\cup }_{\alpha }{e}^{k+1},Y\right)$ has the rational homotopy type of the product space ${\mathcal{ℱ}}_{\ast }\left(X,Y\right)\times {\Omega }^{k+1}Y$. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex $X$ are greater than $WL\left(Y\right)$ and the connectivity of $Y$ is greater than or equal to $dimX$, then the mapping space ${\mathcal{ℱ}}_{\ast }\left(X,Y\right)$ can be decomposed rationally as the product of iterated loop spaces.